Anisotropic Radial Basis Function Methodsfor Continental Size Ice Sheet Simulations

Gong Cheng∗ and Victor Shcherbakov†

November 29, 2017

Abstract

In this paper we develop and implement anisotropic radial ba-sis function methods for simulating the dynamics of ice sheets andglaciers. We test the methods on two problems: the well-known bench-mark ISMIP-HOM B that corresponds to a glacier size ice and a syn-thetic ice sheet whose geometry is inspired by the EISMINT bench-mark that corresponds to a continental size ice sheet. We illustratethe advantages of the radial basis function methods over a standardfinite element method. We also show how the use of anisotropic radialbasis functions allows for accurate simulation of the velocities on alarge ice sheet, which was not possible with standard isotropic radialbasis function methods due to a large aspect ratio between the icelength and the ice thickness. Additionally, we implement a partitionof unity method in order to improve the computational efficiency ofthe radial basis function methods.

1 Introduction

There is a growing interest in simulating the evolution of ice sheets for pre-dicting their contribution to the future sea level rise [1] and also under-standing the processes of forming past landscapes. Mathematical models areintroduced as tools to understand the dynamics of ice sheets in the past andin the future [2]. Improvement in accuracy and efficiency of the modelingand numerical methods is always needed, especially for large scale and longtime simulations [3, 4, 5].

∗Uppsala University, Division of Scientific Computing, cheng.gong@it.uu.se†Uppsala University, Division of Scientific Computing, victor.shcherbakov@it.uu.se

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The ice flow is generally described as an incompressible, non-Newtonianfluid with highly nonlinear viscosity. One accurate model for simulating theice dynamics is the so-called full Stokes equations [6, 7, 8]. The deformationof the ice body under its own weight is governed by Glen’s flow law [9]. Itrelates the stress field to the strain rates as a viscous fluid and the viscositydepends nonlinearly on the velocities, which introduces an additional degreeof difficulty in the numerical solving procedure for the full-Stokes equations.Moreover, the discretisation of the full-Stokes system gives rise to a saddle-point problem, which in a setting for finite element methods requires specialnumerical treatments for the methods to satisfy the inf-sup condition [10],such as adding stabilization on the pressure variables [11] or using high-order finite element methods [12]. Therefore, the numerical solution of thefull-Stokes equations is demanding in terms of computational time.

Several simplifications are derived for the full-Stokes equations to reducethe computational complexity. The first order Stokes model (also knownas the Blatter–Pattyn model) is based on the assumption that the hydro-static pressure is balanced by the vertical normal stress [13, 14], such thatthe horizontal gradient of the vertical velocity is neglected. The system issimplified to an elliptic problem that only contains the horizontal velocitiesas unknowns and the vertical velocity is recovered by solving the incompress-ibility equation. Other approximations are the Shallow Ice Approximation(SIA) [15] and Shallow Shelf Approximation (SSA) [16, 17], which howevergive a lower order of approximation than the first order Stokes model, i.e.,have a larger model error.

These models are intercompared within several benchmark experimentsduring the past decade. For instance, some of the well-known benchmarkexperiments are the Ice Sheet Model Intercomparison Project for Higher-Order ice sheet Models (ISMIP-HOM) [4] and the framework of EuropeanIce Sheet Modeling Initiative (EISMINT) [18]. In ISMIP-HOM, differentStokes approximations are compared on glacier size problems (about 10 kmlong), whereas in EISMINT, the computational domains are continental size(more than 1000 km long).

Traditional numerical methods such as finite element methods (FEM)are commonly used for solving ice sheet models since FEM can easily handlecomplex geometry with different types of boundary conditions. However,it has some drawbacks when solving problems on a domain with a movingboundary, which leads to remeshing of the entire domain for every time step.Also, solving the nonlinear system requires a full matrix reassembly duringeach nonlinear iteration. Therefore, a mesh-free method that can be stated instrong form is considered to be a preferred choice for such problems. Radialbasis function (RBF) methods are of that kind [19]. The idea is to define a

2

finite-dimensional basis, which consists of functions, whose values depend onthe distance from their centers, and use them for approximating the solution.

RBF methods have been introduced for ice simulations in [20], where theauthors have studied the advantages of an RBF approach for the Haut glacierd’Arolla, which is also a test case from the ISMIP-HOM benchmark. In thispaper, we continue the work and extend the approach to solve for dynam-ics of continental size ice sheets. We introduce anisotropic RBF approxima-tions to solve problems with continental ice sheet geometries, which typicallyhave large aspect ratios. The high aspect ratio is an obstacle for standardisotropic RBFs due to the strong dependence of the shape parameter, whichdetermines the width of the functions. The use of anisotropic RBFs signif-icantly relaxes this dependence and simplifies the method implementation.Additionally, we suggest a strategy of selecting the shape parameter valuebased on the conditioning of the interpolation matrix. In order to enablemore efficient simulations we develop a partition of unity approach that isalso based on anisotropic patches. Also, we study error estimates for theanisotropic method and perform convergence tests.

The remainder of the paper is structured as follows. In Section 2 weexplain the full-Stokes equations and the first order Stokes model that gov-erns the dynamics of the ice sheets. Then, in Section 3 we introduce theanisotropic RBF methods, and in Section 4, we present the numerical resultsobtained by the RBF methods for the two test cases. We also provide acomparison of the RBF methods with the standard FEM in terms of time-to-accuracy, i.e., we compare the execution times to achieve certain errorlevels. Finally, we draw some conclusions in Section 5.

2 Ice Sheet Dynamics

Ice can be viewed as a very viscous fluid flow on a large scale. Thus, themodels for simulating ice sheet dynamics are inspired by fluid dynamics laws.In general, ice is considered as an incompressible flow with a low Reynoldsnumber and with the stress tensor related to the strain rate by a power lawviscous rheology [6]. Due to the slow motion of ice masses, the accelerationterm can be neglected and the Navier–Stokes equations can be turned intothe full-Stokes equations that describes a steady flow. However, the computa-tional demand for solving the full-Stokes system can be excessive, especiallyfor such large domains as ice sheets. Therefore, under assumptions of largeaspect ratios between the ice length and ice thickness, the horizontal vari-ations of the vertical velocity in the full-Stokes equations can be neglected.Thereby, we arrive at the simplified first order Stokes equations, which is an

3

approximation to the full-Stokes equations in terms of the thickness/lengthratio. The solution obtained by the first order Stokes equations is accurateat the ice margins, but the areas with discontinuities in the surface gradi-ent might not be resolved well enough, if compared with the solution of thefull-Stokes system.

as

∗∗

∗

∗

∗∗ ∗

∗∗∗

∗

∗

∗∗

∗

∗ ∗∗

∗

∗

∗∗

∗ ∗ ∗∗

∗

∗

∗ ∗h(x)

b(x)

z

x

ns

Figure 1: A schematic continental size ice cap. For aesthetic reasons thecoordinates are exaggerated in the vertical direction.

2.1 The Full-Stokes Equations

The nonlinear full-Stokes equations are defined by the conservation of mo-mentum and mass

∇ ·(η(∇v +∇vT )

)−∇p+ ρg = 0,

∇ · v = 0,(2.1)

where v is the vector of velocities v =(vx vy vz

)T, ρ is the density of

the ice, η is the viscosity, p is the pressure, and g is the gravitational ac-celeration. A constitutive equation (so-called Glen’s flow law) relates thedeviatoric stress tensor TD and the strain rate D = 1

2

(∇v + (∇v)T

)such

thatD = A(T ′)f(σ)TD, (2.2)

where A(T ′) is the rate factor that describes how the viscosity depends onthe pressure melting point corrected temperature T ′. For isothermal flow,the rate factor A is constant. The value of the physical parameters are givenin Table 1.

4

The deviatoric stress tensor TD is given by

TD =

tDxx tDxy tDxztDyx tDyy tDyztDzx tDzy tDzz

, (2.3)

where tDxx, tDyy, t

Dzz and tDxy denote longitudinal stresses and tDxz, t

Dyz vertical

shear stresses. The function f(σ) is called the creep response function for icewith

f(σ) = σn−1, (2.4)

where n is the flow-law exponent factor and σ is the effective stress, which isdefined as the second invariant of the deviatoric stress tensor

σ =

[(tDxy)2

+(tDyz)2

+(tDxz)2

+1

2

((tDxx)2

+(tDyy)2

+(tDzz)2)] 1

2

. (2.5)

The effective strain rate is defined in analogy with the effective stress suchthat

De =

√1

2tr(D ·D

), (2.6)

with the relationDe = Aσn, (2.7)

and the viscosity η is defined by

η =(2Af(σ)

)−1=

1

2A− 1

nD1−nn

e . (2.8)

At the top surface with normal ns, the ice is stress-free, i.e.,(− pI + 2ηD

)· ns = 0, (2.9)

where I is the identity matrix.At the base of the ice, the normal vector nb and the tangential vectors t1

and t2 span the base surface such that nb · ti = 0, i = 1, 2 and t1 · t2 = 0. Ifthe ice base is frozen, the velocity v satisfies a no slip condition at the base

v = 0. (2.10)

The time evolution of the domain geometry is introduced by a kinematicboundary condition at the top surface, which moves the surface h(x, y) ac-cording to the surface velocities and the accumulation-ablation function assuch that

∂h

∂t+ vx

∂h

∂x+ vy

∂h

∂y= as + vz, (2.11)

where as is in the unit of meters per annum (m/a) ice equivalent. Positivevalues of as imply snowing and negative values imply snow melting.

5

Parameter Value Unit Descriptionρ 900 kg m−3 Ice densityg 9.81 m s−2 Acceleration of gravityn 3 – Flow-law exponentA 10−16 Pa−3a−1 Rate factor in flow law

Table 1: The physical parameters of ice.

2.2 The First Order Stokes Model

As the full-Stokes system consists of complex equations with a highly non-linear viscosity, they require a massive computational effort to be solvednumerically. Moreover, the system gives rise to a saddle point problem thatadds an extra degree of difficulty. Therefore, the full-Stokes equations areoften simplified to a reduced form under the assumption that the variationalstress is neglected [21] due to the large aspect ratio. Thus, the conservationof momentum in (2.1) becomes

∂

∂x

(4η∂vx∂x

+ 2η∂vy∂y

)+

∂

∂y

(η∂vx∂y

+ η∂vy∂x

)+

∂

∂z

(η∂vx∂z

)= ρg

∂h

∂x,

∂

∂x

(η∂vx∂y

+ η∂vy∂x

)+

∂

∂y

(4η∂vy∂y

+ 2η∂vx∂x

)+

∂

∂z

(η∂vy∂z

)= ρg

∂h

∂y,

(2.12)

where h(x, y) denotes the surface elevation and the viscosity is written as

η =1

2A− 1

n

[(∂vx∂x

)2

+

(∂vy∂y

)2

+∂vx∂x

∂vy∂y

+1

4

(∂vx∂y

+∂vy∂x

)2

+1

4

(∂vx∂z

)2

+1

4

(∂vy∂z

)2] 1−n

2n

.

(2.13)

The vertical velocity is obtained through vertical integration over the con-servation of mass in (2.1) from the bottom b(x, y) to a height z(x, y)

vz(z)− vz(b) = −∫ z

b

(∂vx∂x

+∂vy∂y

)dξ. (2.14)

6

The stress-free boundary on the top surface is expressed in terms of thevelocity gradients such that(

4η∂vx∂x

+ 2η∂vy∂y

)∂h

∂x+

(η∂vx∂y

+ η∂vy∂x

)∂h

∂y− η∂vx

∂z= 0,(

4η∂vy∂y

+ 2η∂vx∂x

)∂h

∂y+

(η∂vx∂y

+ η∂vy∂x

)∂h

∂x− η∂vy

∂z= 0.

(2.15)

The system of equations that has to be solved is referred to as the firstorder Stokes or the Blatter–Pattyn model. It is second order accurate withrespect to the thickness/length ratio. The computational demand for solvingthe Blatter–Pattyn model is significantly lower than for solving the full-Stokessystem. Since in this paper we are not dealing with simulations of groundinglines and calving fronts, we use the Blatter–Pattyn model which gives asatisfactory solution.

2.3 The Flow-Line Model

We consider the two-dimensional flow-line model, which is a vertical cuttingplane along the surface gradient. This is a simplification from the three-dimensional ice sheet to a two-dimensional problem, which is commonly usedin glaciology. The Blatter–Pattyn equations in (2.12) combined with theconservation of mass in (2.1) are reduced to

4∂

∂x

(η∂vx∂x

)+

∂

∂z

(η∂vx∂z

)= ρg

∂h

∂x,

∂vz∂z

= −∂vx∂x

,

(2.16)

with the viscosity (using n = 3 in Table 1)

η =1

2A−1/3

[(∂vx∂x

)2

+1

4

(∂vx∂z

)2]−1/3

. (2.17)

The horizontal velocity vx is determined by the conservation of momentumin (2.16) and the vertical velocity vz can be computed by vertically integrat-ing vx from the bottom of the ice to the position z as in (2.14)

vz(z)− vz(b) = −∫ z

b

∂vx∂x

dz. (2.18)

The geometry of the ice bottom is constrained by the bedrock and thebedrock is assumed to have no deformation during the whole simulation.

7

There is no sliding on the bedrock, therefore the horizontal and verticalvelocity at z = b are both equal to zero. On the top surface, the stress-freeboundary condition is simplified as

η

(4∂vx∂x

∂h

∂x− ∂vx

∂z

)= 0. (2.19)

The lateral boundary conditions are given in the following ways for differenttest cases:

1. For the periodic boundary conditions, e.g., in the ISMIP-HOM bench-mark experiment, the solution from the right boundary is mapped tothe left boundary.

2. At the far end of the thin ice area, e.g., the left and right boundariesin the ice cap experiment, a symmetric boundary condition is imposedas

∂h

∂x= 0, vx = 0. (2.20)

3 Radial Basis Function Methods

In order to solve the system (2.16) numerically, we develop and implementRBF methods. To construct an RBF approximation we scatter a set ofnodes over the computational domain, where each node is associated witha basis function. The collection of basis functions forms a finite basis inthe functional space. Some typical choices of basis function can be foundin Table 2. An important feature of an RBF is that it depends only onthe distance between the nodes and its center. This is a valuable propertysince it makes the method easily applicable to high dimensional problems.Additionally, RBF methods are mesh-free and therefore suitable for problemswhich are defined in domains with complex geometries, such as ice sheets andglaciers.

Table 2: Commonly used smooth radial basis functions.

RBF φ(ε, r)

Multiquadric (MQ) (1 + (εr)2)1/2

Inverse Multiquadric (IMQ) (1 + (εr)2)−1/2

Inverse Quadratic (IQ) (1 + (εr)2)−1

Gaussian (GA) e−(εr)2

8

Given N distinct scattered nodes x = x1, . . . , xN, xi ∈ Ω ⊂ Rd, wecan construct an RBF approximation v(x) of a function v(x) with valuesv = [v(x1), . . . , v(xN)] defined at the nodes such that

v(x) =N∑j=1

λjφ(ε, ||x− xj||), x ∈ Ω, (3.1)

where λj are the unknown coefficients, ‖ ·‖ is the Euclidean norm and φ(ε, r)is a real-valued radial basis function, whose width is determined by the shapeparameter ε. In order to determine the coefficients λj we collocate the ap-proximation (3.1) at the node set and obtain a system of linear equations forthe coefficients λ

Aλ = v, (3.2)

where the matrix A has the following elements Aij = φ(ε, ‖xi − xj‖).

3.1 Anisotropic Radial Basis Functions

The thickness of a continental ice sheet is relatively small in comparison withits length. The ratio between the length and thickness may in some casesreach 500 : 1. Therefore, approximation with standard RBFs fails to providea reasonable resolution in the vertical direction. Several authors suggestedto use anisotropic (elliptic) RBFs instead [22, 23, 24]. These are functions,which are scaled in an appropriate way to give good resolution and matchthe domain geometry features (see Figure 2). This can be implemented byredefining the distance between two nodes such that

‖x− y‖a =√a2

1(x1 − y1)2 + · · ·+ a2d(xd − yd)2, x, y ∈ Ω ⊂ Rd, (3.3)

where ai are the aspect ratios of the domain discretisation defined as the ratiobetween typical node distances in the respective directions (see Section 4 foran accurate definition). Without loss of generality, we assume a1 = 1 andai ≥ 1, i = 2, . . . , d. Particularly, for the two-dimensional problems that weconsider in this paper the vertical dimension is several orders of magnitudesmaller than the horizontal dimension. Therefore, the norm can be definedas

‖x− y‖a =√

(x1 − y1)2 + a2(x2 − y2)2, x, y ∈ Ω ⊂ R2. (3.4)

Thus, the basis function φ is no longer a function of the Euclidean distancebut a function of the scaled distance defined by the ‖ · ‖a-norm.

9

Figure 2: Top: An anisotropic Gaussian RBF. Bottom: An isotropic Gaus-sian RBF. The aspect ratio a = 10.

3.2 Kansa’s Method

The RBF interpolant (3.1) can also be collocated for a partial differentialequation (PDE) or a system of partial differential equations. This approachis known as Kansa’s method [25, 26]. The nonlinear First Order StokesEquations (2.16) can be written in general form as a nonlinear boundaryvalue problem

P [x, v(x),Dv(x)] = 0 ⇐⇒P1 = 0, x ∈ Ω,

P2 = 0, x ∈ ∂Ω,(3.5)

where P1 is the interior nonlinear operator, P2 is the boundary nonlinearoperator, and D is a shorthand notation for differential operators, such as∂x, ∂z, ∇.

A root of the nonlinear system (3.5) can be sought by a nonlinear solver,such as Newton’s method [27] or a fixed point iteration method [20], thatiteratively solves a linearised problem. Thus, we arrive at a system of linearequations in the following form

Lv(x) = f(x), x ∈ Ω,

Fv(x) = g(x), x ∈ ∂Ω,(3.6)

where L is the linearised interior differential operator, F is the linearisedboundary differential operator, and f , g are the right hand side forcing func-tions. We seek a solution to system (3.6) in the form of the RBF inter-polant (3.1). Collocating at the node points we obtain the following systemof linear equations

Cλ :=

[LF

]λ :=

[LII LIBFBI FBB

] [λIλB

]=

[fI

gB

], (3.7)

10

where L, F , f , and g are discrete representations of the continuous quantities,and the subscripts I and B denote that the quantities are evaluated on theinterior and the boundary nodes, respectively (without loss of generality,we assume that the first NI nodes belong to the interior of Ω and the lastNB = N − NI nodes belong to the boundary ∂Ω). The matrices L andF are constructed from the elements Lij = Lφ(ε, ‖xi − xj‖a) and Fij =Fφ(ε, ‖xi − xj‖a).

It has been shown [28, 29, 30] that for smooth RBFs the magnitude ofthe coefficients becomes unbounded as ε → 0, while the values v(x) remainwell-behaved. Therefore, we prefer to transform the problem into a searchfor the nodal values v(x). For the basis functions presented in Table 2 theinterpolation matrix A is non-singular for distinct node points [31]. Hence,based on (3.2) we can write

λ = A−1v. (3.8)

Thus, relation (3.8) allows us to transform the problem from solving for λ todirectly solving for v.[

LII LIBFBI FBB

] [λIλB

]=

[LII LIBFBI FBB

]A−1

[vIvB

]=

[fI

gB

]. (3.9)

3.3 Radial Basis Function Partition of Unity Method

The approach presented in the previous section is referred to as a global RBFapproximation, since it is constructed over all discretisation nodes. Such anapproach gives a highly accurate approximation, but results in a dense systemof linear equations, which is computationally expensive to solve. To overcomethis issue we employ a partition of unity method that allows for a significantsparsification of the linear system. Thereby, the high computational costassociated with the global method is reduced, while a similarly high accuracyis maintained [20, 32]. Moreover, a partition based formulation is well suitedfor parallel implementations.

The partition of unity method was first introduced for finite elementmethods by Babuska and Melenk in [33]. Later it was applied to RBF-based formulations by multiple authors [19, 32, 34, 35, 36]. The main idea ofthe method is to subdivide the computational domain into subdomains andconstruct an RBF interpolant locally in each subdomain and then combinethem together by the partition of unity functions, which serve as weights.Below comes a formal description of the method.

To define a partition of unity method for problem (3.6) we construct aset of overlapping patches ΩkMk=1 that form an open cover of the domain Ω,

11

such that

Ω ⊂M⋃k=1

Ωk. (3.10)

The patches Ωk are selected as circular discs in the anisotropic norm || · ||a.Additionally, we construct a partition of unity wkMk=1 that is subordinatedto the open cover ΩkMk=1. The function wk is compactly supported on Ωk,and

M∑k=1

wk(x) = 1, x ∈ Ω. (3.11)

Thus, the RBF approximation can be written in the form of a weighted sumof all local approximations

v(x) =M∑k=1

wk(x)vk(x), x ∈ Ω, (3.12)

where vk(x) is a local approximation defined as

vk(x) =

nk∑i=1

λki φ(ε, ‖x− xki ‖a), x ∈ Ωk, (3.13)

and nk is the local number of computational nodes in the patch Ωk. The parti-tion of unity weight functions can be constructed using Shepard’s method [37]

wk(x) =ϕk(x)∑Mi=1 ϕi(x)

, k = 1, . . . ,M. (3.14)

In order to provide necessary smoothness of the solution we choose ϕk(x) asa C2(R2) compactly supported Wendland function [38]

ϕ(r) =

(1− r)4(4r + 1), if 0 ≤ r ≤ 1,

0, if r > 1,(3.15)

which is scaled and shifted accordingly to fit the partition Ωk with the centreck and radius ρk

ϕk(x) = ϕk

( ||x− ck||aρk

), ∀x ∈ Ω. (3.16)

An example of the domain partitioning is shown in Figure 3 with anice cap geometry spanning [0, 1500]× [0, 3.5] kilometres. We use anisotropicpartitions in this application as it was done in [35]. The black ellipses are

12

Figure 3: Top: An example of an anisotropic partitioning over the ice capdomain. Bottom left: Zoom-in on one partition. Bottom right: The sparsestructure of the interpolation matrix for the ice cap domain with the dis-cretization and partitioning as on the top panel.

the boundaries of the partitions centered at the red dots and the green dotsare the computational nodes in the background node set. A zoomed-in plotof one partition is shown on the bottom left panel in Figure 3. The ratiobetween the major and the minor axis of the partition is equivalent to theaspect ratio a. Consequently, the partitions can be considered as circularlyshaped under the || · ||a-norm given in (3.4). On the bottom right panel,we show the sparsity structure of the interpolation matrix for the ice cap on2311 Cartesian nodes with the partitioning as in the top panel. There are18.9% nonzero element in the matrix with an average bandwidth around 800elements. The bandwidth will not grow with the number of computationalnodes as long as the partition size is fixed to contain a certain number ofnodes, e.g., for the finest grid that was used in our experiments for the icecap test case the coefficient matrix had only 2.3% of nonzero elements.

Note that the amount of overlap between patches is limited in order to

13

maintain a sparse pattern of the linear system. Greater overlap leads to aslightly higher accuracy but denser linear system and, hence, higher compu-tation costs. Less overlap leads to a lower accuracy but at the same time toa lower computational cost. We find that 20%− 30% overlap is appropriate.More on this matter can be found in [32].

3.4 Error Estimates

We are interested in estimating the difference between the exact unknownsolution v and its RBF approximation v. We define the PDE error as

E = v − v. (3.17)

We assume that we solve the already linearized problem (3.6) and the errorintroduced by the linearization is negligible in comparison with the approx-imation error. Error estimates for RBF approximations of PDE solutionswere studied by several authors [35, 39] in cases when isotropic Gaussian andinverse multiquadric basis functions are used. They demonstrated that thePDE error is governed by the RBF interpolation error

EL = L(v − J (v)

). (3.18)

Here J (v) is the RBF-PUM interpolant defined as

J (v) =M∑k=1

wkJ (vk), (3.19)

where J (vk) is the local RBF interpolant defined as in (3.13) satisfying theinterpolation condition

J (vk)(xk) = v(xk), (3.20)

where xk are the nk local nodes that belong to the patch Ωk. Therefore, in thissection we aim to adapt the existing isotropic interpolation error estimatesfor the anisotropic Gaussian RBFs.

In particular, the authors of [35, 40, 39] consider two node refinementstrategies: (i) refining the node set while having the domain partitioningfixed; and (ii) refining the node set and simultaneously refining the domainpartitioning. Strategy (i) leads to an increasing number of local nodes in eachpatch and, thereby, yields an exponential convergence (for smooth problems).The limitation of this strategy is that after just a few iterations of noderefinement the coefficient matrix becomes highly ill-conditioned. Strategy (ii)

14

leads to an increasing number of patches that contain a similar number oflocal nodes in every iteration of node refinement. This strategy yields onlyan algebraic convergence, but the condition number of the coefficient matrixdoes not increase as rapidly. In this paper we are interested in and adhereto Strategy (ii) because it allows for larger scale simulations.

Thus, recapitulating the results from [35, 39] for isotropic RBF-PUM wehave

||EL||L∞(Ω) ≤ K max1≤k≤M

∑|β|≤|α|

(α

β

)||Dβwk||L∞(Ωk)||Dα−β(vk − J (vk)

)||L∞(Ωk),

(3.21)where Ωk = Ω ∩ Ωk, vk is the global solution restricted to Ωk, α is the de-gree of the differential operator L, defined as a multi-index α = (α1, . . . , αd),Dα is the differentiation operator of degree α, and K is some constant. Thepartition of unity weight functions wk are required to have bounded deriva-tives up to order α, i.e.,

||Dαwk||L∞(Ωk) ≤Cα

H|α|k

, (3.22)

whereHk is the diameter of Ωk. From (3.21) using the results of [40] combinedwith (3.22) one can obtain the following error estimate, which indicates analgebraic rate of convergence,

||EL||L∞(Ω) ≤ K max1≤j≤M

CAk H

q(nk)+1− d2−|α|

k ||v||N (Ωk), (3.23)

where the norm || · ||N (Ωk) denotes the native space norm (see [19, 40]) associ-ated with the chosen RBF, the function q(nk) corresponds to the polynomialdegree q supported by the local number of points nk, d is the dimensionalityof the problem, and CA

k are some constants that depend on the dimension-ality d, the chosen weight function, the number of local points nk, and theorder of the differential operator.

Introducing anisotropic scaling is equivalent to changing the distancefunction, in which RBFs are defined. The properties of the functions re-main unchanged. Differentiation of an anisotropic RBF gives rise to theaspect ratio constant coming from the distance defined in || · ||a. The sameapplies to differentiation of the partition of unity weight functions. Hence,we can adapt (3.21) in the following way

||EL||L∞(Ω) ≤ K max1≤j≤M

∑|β|≤|α|

(α

β

)aβ||Dβwk||L∞(Ωk)×

aα−β||Dα−β(vk − J (vk))||L∞(Ωk). (3.24)

15

whereaβ = aβ11 × . . .× aβdd . (3.25)

That is, in total we get the factor aα appearing in the inequality. Thereby,we can again use the results of [40] to obtain the following error estimatethat adapts for the anisotropic RBF-PUM

||EL||L∞(Ω) ≤ aαK max1≤j≤M

CAk H

q(nk)+1− d2−|α|

k ||v||N (Ωk). (3.26)

The relation (3.26) goes in line with the results for error bounds for theinterpolation by anisotropic RBFs obtained in [22]. Also, it indicates thatthe approximation by the anisotropic RBF-PUM converges algebraically ifthe refinement strategy (ii) is employed keeping the norm fixed.

4 Numerical Experiments

In this section we present results of numerical experiments for the ISMIP-HOM B benchmark test and for the two-dimensional ice cap. The ISMIP-HOM B test is a glacier size problem, whereas the two-dimensional ice caprepresents a continental size ice sheet. We illustrate the advantages of theanisotropic radial basis function methods over the standard finite elementmethod (with piecewise linear basis functions and anisotropic unstructuredmesh) on the ISMIP-HOM B test. In the two-dimensional ice cap simulation,we demonstrate the capability of the anisotropic RBF partition of unitymethod to solve a continental size ice sheet problem, where the ratio betweenthe ice length and height is around 428 : 1. We would like to emphasizehere that isotropic RBF methods fail to provide any adequate solution. Thereason is the impossibility to select a suitable shape parameter that woulddefine basis functions of the appropriate shape to properly resolve in bothdirections. Preparing this paper we tested isotropic RBF methods on severaldifferent node layouts with various shape parameters. However, in none of thetests we succeeded to obtained a stable solution. Here we do not provide thevelocity fields approximated by isotropic RBF methods because they makeno sense. Potentially, one could use the same node set resolution in both x-and z-directions, but then having, say, 35 nodes in the vertical direction withthe maximal ice thickness of 3.5 km would require having 15000 nodes in thehorizontal direction to provide the same node set resolution, if the ice lengthis 1500 km. Such a resolution would lead to a high computational cost ofthe algorithm. Therefore, we are certain that anisotropic RBF methods aremuch more suitable for approximation of the dynamics of continental size icesheets.

16

In order to construct a set of computational nodes we use (i) a back-ground Cartesian grid and (ii) a set of Halton nodes. The Cartesian gridis characterized by Nx and Nz nodes in the x- and z-directions with res-olutions hx and hz, respectively, which are distributed over the rectangle[0, Lx]× [0, Lz], where Lx is the length of the domain and Lz is the maximumthickness. The quasi-random Halton nodes are also selected to fill up therectangle [0, Lx]× [0, Lz] with Nx×Nz nodes. To define the resolution of theHalton node set we simply introduce the notation (likewise for the Cartesiangrid) hx = Lx/(Nx − 1) and hz = Lz/(Nz − 1). After defining nodes for therectangle, the nodes which fall outside the domain profile are discarded, andthe ones which fall inside are kept and augmented by the boundary nodes,thereby resulting in a total of N computational nodes.

The Cartesian background grid is very convenient and easy to construct.However, it is not always the most suitable node layout for the RBF approxi-mation [41] but allows to select the shape parameter ε relatively simply sincethe distance in both x- and z-directions is uniform in the || · ||a-norm anddoes not vary. In contrast, the Halton nodes might be a bit more challengingto select a proper shaper parameter for. Although, the Halton nodes havea good space filling property and are, thus, maybe more suitable for largesimulations.

So far no analytical procedures of finding the shape parameter have beendeveloped. Therefore, we seek the shape parameter empirically in the follow-ing form

ε =C

h, (4.1)

where C is a constant and h is the internodal distance defined in the a-normas

h =√h2x + a2h2

z. (4.2)

and the aspect ratio a is defined as the ratio between the spacings in the x-and z-directions.

a =hxhz. (4.3)

We find that C = 0.5 is a fairly good choice for the ISMIP-HOM B benchmarktest and that C = 0.125 is a suitable choice for the ice cap test case. In orderto get an intuition on how to choose the value of the constant C we adopt astrategy of controlling the condition number of the interpolation matrix A (incase of partition of unity this is the global interpolation matrix reassembledfrom the local interpolation matrices with the partition of unity weights). Wemaintain the condition number around 1016, i.e., around near ill-conditioning,as it is a well-known fact that the RBF approximation achieves its best

17

accuracy for small values of ε > 0 [29, 42, 43], which leads to a large conditionnumber of the matrix. We repeat this procedure several times on coarsenode sets and record the condition numbers of the resulting matrices. Thenwe utilize a least squares fit to determine the value of C and reuse thisvalue for finer node sets. To estimate the condition number we use the1-norm condition estimator of Hager [44] and a block-oriented generalizationof Hager’s estimator [45] provided by the function condest in MATLAB.Some other strategies and insights on how to determine a proper value of theshape parameter can be found in [20, 46].

4.1 ISMIP-HOM B

We compute the velocity field of an ice slab that is grounded on a bedrockthat has sinusoidal shape. The length of the slab is L = 10 km. The surfaceis defined as a function of the x-coordinate

hs(x) = −x tanα, x ∈ [0, L], (4.4)

and has a slope α = 0.5. The bedrock is defined as

hb(x) = hs(x)− 1000 + 500 sin

(2πx

L

), x ∈ [0, L]. (4.5)

Figure 4: Top: The horizontal velocity for the ISMIP-HOM B test computedby the global RBF method (left) and FEM (right). Bottom: The verticalvelocity computed by the global RBF method (left) and FEM (right). Allvelocities are measured in km/a.

We assign periodical boundary conditions at the lateral boundaries. Also,we assume that the ice is grounded on the bedrock, meaning no slip boundary

18

condition on the bottom, and the ice surface is free of stress, which leads toa free surface problem.

The horizontal and vertical velocities, obtained by the global RBF methodand FEM on node sets with similar resolution, are presented in Figure 4.1.The two left panels represent the RBF solutions, while the right panels rep-resent the FEM solutions, which we use for comparison. The horizontal andthe vertical velocities are well resolved by both methods.

Table 3: Maximum absolute errors (m/a) in the horizontal velocity and CPUtimes for several different numbers of degrees of freedom for FEM, global RBFmethod, and RBF-PUM for the ISMIP-HOM B benchmark test.

FEM RBF RBF-PUM

Error N Time (s) Error N Time (s) Error N Time (s)0.3353 243 2.2782 0.2535 306 0.2388 0.3467 291 0.24790.1528 805 8.0495 0.1574 802 1.3701 0.1365 771 0.76850.0492 2889 34.5913 0.0432 1992 13.7930 0.0372 1925 4.1405

We compare three methods: the global RBF method, RBF-PUM, andFEM in terms of accuracy and computational efficiency. We use a finiteelement solution on a fine grid with 10897 degrees of freedom as our referencesolution. The maximum absolute errors, run times, and numbers of degreesof freedom are presented in Table 3. We observe that the RBF methodsgive better accuracy in much shorter time. RBF-PUM is more than 8 timesfaster than the standard FEM. Additionally, thanks to the high accuracy,the RBF methods need fewer nodes to reach a similar error tolerance level.This is crucial for large simulations since less storage will be required. Thereason that the FEM implementation runs slower is the necessity of matrixreassembly within the nonlinear iteration. It has been shown [47] that thisprocedure may severely dominate the total computational time.

Thereby, the numerical results for the ISMIP-HOM B demonstrate thatRBF-PUM is the most efficient out of the three methods for this type ofproblem. Inspired by this, we continue investigating the properties of theanisotropic RBF-PUM. However, in the remaining part of the paper we willnot provide comparisons with the global RBF method and FEM, because theglobal RBF method becomes too computationally intensive, while a rigorousimplementation of FEM for the two-dimensional ice cap goes beyond thescope of this paper.

19

4.2 The Two-Dimensional Ice Cap

This experiment is inspired by the EISMINT benchmark test [18], wherenumerical methods are intercompared on a continental size ice sheet. Thereare several challenges in simulating ice dynamics with RBF methods even atsteady states, for instance, a large aspect ratio of the computational domainor a dramatic variation of the ice thickness from the margins to the centreof the cap. These factors would require a delicate adjustment of the shapeparameter in the isotropic RBF case, but it is much simpler for the anisotropicRBFs since the issues with the aspect ratio can be resolved by introducingthe distance in the || · ||a-norm. Thus, in this test problem we would like toillustrate the clear advantages of anisotropic RBF methods over the standardisotropic RBF methods.

The computational domain follows the so-called Bueler profile [6, 48],which is a two-dimensional flow-line model spanning from x = 0 km tox = 1500 km. At steady state, the ice cap covers the area from x = 300 kmto x = 1200 km and the “ice free” area is at x ∈ [0, 300] and x ∈ [1200, 1500],(see Figure 3). Generally, in order to make it possible for the margin of theice cap to extend, a thin layer (10 m) of ice is assumed on the “ice free” area.

The physical parameters of this experiment are given in Table 1. There isno sliding on the bedrock and no velocity on the lateral boundaries. The topsurface is free of stress by neglecting the atmospheric pressure. In order toaccurately capture the surface gradient Chebyshev points are used to generatethe upper boundary discretisation on the ice cap part. The “ice free” regionsand the bottom boundaries are discretised using equidistant nodes.

4.2.1 The Cartesian Nodes

The anisotropic RBF provides the flexibility to choose any combination ofNx and Nz. However, we need to keep in mind that the expression (4.3) forthe aspect ratio relates Nx and Nz as

Nx

Nz

=LxaLz

. (4.6)

Here the aspect ratio can be considered as a scaling of the z-direction. Ideally,we prefer to have the final geometry after the scaling close to a square. Thatis, we need to select Nx and Nz with some special care. Therefore, in thisapplication we require the numbers Nx and Nz to be of the same order ofmagnitude.

Also, note that when combining the background and boundary nodes weneed to account for nearly indistinct nodes, i.e., nodes positioned in a tiny

20

neighbourhood of each other, since this can lead to a highly ill-conditioned orsingular coefficient matrix. In order to avoid the ill-conditioning we removethe internal nodes whose distance to the boundary is shorter than h/4.

0 500 1000 1500

0

1

2

3

-0.2

0

0.2

0 500 1000 1500

0

1

2

3

-10

-5

0

5

10-3

Figure 5: Top: The horizontal velocity (in km/a) of the ice cap computed onCartesian nodes by anisotropic RBF-PUM. Bottom: The respective verticalvelocity (in km/a).

The result of a numerical simulation with anisotropic RBF-PUM is pre-sented in Figure 5. The horizontal velocity is displayed in the upper paneland the vertical velocity is displayed in the lower panel. The unit of thevelocities is kilometre per annum (km/a). The background grid is charac-terised by Nx = 60 and Nz = 35, yielding N = 1014 computational nodesin the domain. The reader is referred to [18] to ascertain that the presentedsolutions are qualitatively correct.

4.2.2 The Halton Nodes

Another valuable property of RBF methods is their mesh-free nature, whichmakes them suitable for problems set in domains with complex geometries.Moreover, having the mesh-free property we can cluster computational nodesand achieve a higher resolution in the regions where it is required, e.g., alongthe margins. In order to demonstrate the capability of anisotropic RBFmethods to work on unstructured node sets we select the quasi-random Hal-

21

ton node layout. The scaling of the anisotropic basis functions is based onaspect ratio as a function of the mesh resolutions as in relation (4.3).

0 500 1000 1500

0

1

2

3

-0.2

0

0.2

0 500 1000 1500

0

1

2

3

-10

-5

0

5

10-3

Figure 6: Top: The horizontal velocity (in km/a) of the ice cap computed onthe Halton nodes by anisotropic RBF-PUM. Bottom: The respective verticalvelocity (in km/a).

We generate and scatter the Halton nodes over the computational do-main according to the strategy given in the beginning of Section 4. After weaugment them with the boundary nodes and remove the indistinct nodes toavoid singularities in the coefficient matrix. In total we obtain 1063 nodes inthe computational domain, which are used to construct the approxiamtion.In Figure 6 we display the solutions by anisotropic RBF-PUM. Both hori-zontal and vertical velocities are properly resolved and the results go in linewith the results obtained on the Cartesian nodes.

4.2.3 Convergence Tests

In this section, we study convergence of the anisotropic RBF-PUM for bothCartesian and quasi-random nodes. The reference solution is computed bythe anisotropic RBF-PUM on 23943 Cartesian nodes. To generate a conver-gence experiment we repeatedly solve for the horizontal velocity graduallyrefining the node set and measure the root mean square error against the ref-erence solution. The inequality (3.26) holds for the root mean square error

22

since1√N||EL||L2(Ω) ≤ ||EL||L∞(Ω). (4.7)

For the convergence experiments we used a mode with approximately 150nodes per internal patch. However, the patches, which are close to the icemargins, contain just around 20-25 nodes. Therefore, the overall convergencerate was dictated by the order of the local error convergence in those patches.Having the number of local nodes nk = 21 means that the highest polynomialdegree that can be supported in two-dimensional space is q(nk) = 5. That is,taking into account that the problem dimensionality d = 2 and the order ofthe differential operator |α| = 2, we conclude that the anisotropic RBF-PUMapproximation should converge with order 3.

150 200 250 300

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Cartesian

order 2.7

Halton

order 2.1

10001500

20002500

1

2

4

8

Cartesian: Solving

Cartesian: Assembly

Halton: Solving

Halton: Assembly

Figure 7: Left: The convergence of the approximation of the horizontalvelocity obtained by the anisotropic RBF-PUM with respect to the patchsizes H on the Cartesian nodes (blue) and Halton nodes (red) in the log-logscale. Right: The computational time of the matrix assembly and solvingprocedure with respect to the total number of computational nodes in thelog-log scale.

As we can see in the left panel of Figure 7, the rate of convergence on theCartesian nodes with respect to the patch radius is around 2.7, which goesin line with estimate (3.26) as well as with the results in [20]. The rate ofconvergence on the Halton nodes is slightly lower and is around 2.1, whichmakes it require about

√2 times more nodes to reach a similar quality of

23

the solution. Roughly speaking, with a patch resolution of 150 km we getthe horizontal velocity resolved up to 6 m on the Cartesian nodes and up to14 m on the Halton nodes.

-1e2 -1 -1e-2

-2

0

2

-1e3 -1e2 -1e1 -1

-100

-50

0

50

100

-1e-2 -1e-3 -1e-4 -1e-5

-2

0

2

10-4

-1e-1 -1e-2 -1e-3 -1e-4

-5

0

5

10-3

Figure 8: Top: Spectra of the differential operator L discretized by isotropicRBFs (left) and anisotropic (right) RBFs on a coarse node set with 736Cartesian nodes. Bottom: Their counterparts for L discretized on a finenode set with 2562 Cartesian nodes.

In general, we could employ adapted node layouts and patches with thesize matching the domain profile to contain the same number of nodes re-gardless of their position to relax the constraint on the convergence rate,but we leave it to be implemented elsewhere. Also, in general, our experi-ence shows that one should not expect very high convergence rates for thistype of problems. The considered problem is nonlinear with the viscosityinversely proportional to the velocity, meaning that it approaches infinity forlow velocities, which introduces a singularity in the formulation. In order toavoid the singularities and proceed with the numerical solution we preventthe viscosity of growing above 1010 by setting a threshold. However, thisturns the viscosity into a non-smooth function, which usually does not allowfor getting optimal convergence rates.

We present the computational cost of assembling the RBF matrices and

24

solving the nonlinear system in terms of CPU time in Figure 7. The CPUtime is averaged over five independent runs for each resolution to eliminatesystem noise. For both Cartesian and Halton nodes the time for the matrixassembly is a linear function of the total number of nodes since the partitionof unity method provides a sparse structure of the system and the number ofnodes per partition remains fixed such that the bandwidth of the sparse ma-trix does not grow with the total number of nodes. In the solving procedure,we use the default backslash operator in MATLAB. The cost of solving thenonlinear system increases linearly or perhaps slightly super-linearly with thetotal number of computational nodes thanks to the sparse structure of thecoefficient matrix (opposed to O(N3) for dense systems).

Additionally, we look at the spectra of the linearized Blatter–Pattyn op-erator L discretized on two Cartesian node sets (coarser and finer) by bothisotropic and anisotropic RBF-PUM and present them in Figure 8. We usethe same RBF for both methods. The only difference is that the RBF is eval-uated with respect to the Euclidian norm for the isotropic case and with re-spect to the || · ||a-norm for the anisotropic case. For the operator L we wouldlike to see all eigenvalues in the negative half plane, close to the real axis.We observe that both isotropic and anisotropic approximations give spectra,which are rather aligned along the real axis. However, we also notice that theisotropic RBF-PUM yields a spectrum, which is more clustered towards theorigin. In fact, using MATLAB’s eig function for computing matrix eigen-values, we find that there is at least one eigenvalue with <(µ) = 0. That is,the discrete operator is numerically singular. Therefore, the isotropic RBF-PUM is not capable to provide any sensible approximation. In contrast, theanisotropic RBF-PUM yields eigenvalues, which are distinct from the origin,and the ratio max

(|<(µ)|

)/min

(|<(µ)|

)decreases under the node set refine-

ment and is 2 · 105 for a set of 736 Cartesian nodes and 1.7 · 104 for a set of2562 Cartesian nodes.

5 Conclusions

The goal of this work was to extend the approach in [20] for domains withextreme length/thickness ratios to enable approximating the velocities of con-tinental size ice sheets. The main obstacle was the incapability of standard(isotropic) RBF methods to cope with different scales of ice sheet modelling.In order to overcome this issue we developed and implemented an anisotropicRBF partition of unity method that accounts for the aspect ratio of the do-main discretisation and incorporates the ratio into the basis function struc-ture through the updated distance function. This allows us to modify the

25

basis functions in such a way to best fit the domain geometry.The anisotropic RBF-PUM was tested on a glacier size benchmark test

ISMIP-HOM B and on a synthetic continental size ice sheet whose geome-try was inspired by the EISMINT benchmark. The method was comparedwith the global RBF method and FEM for the glacier size test case anddemonstrated a much greater efficiency than its counterparts.

To recover the velocity field of the ice cap we used Cartesian and Haltonnodes. The Cartesian nodes were found to give a better accuracy than theHalton nodes. However, the difference in the results was not critical. In fact,our main idea of implementing the Halton nodes was to demonstrate thatthe anisotropic RBF-PUM is suitable for both structured and unstructurednode sets.

Additionally, we adapted the error estimates from [35, 39] for the anisot-ropic RBF-PUM. We tested the convergence of the solutions obtained bythe anisotropic RBF-PUM on the two node layouts. The convergence rateson the Halton and Cartesian nodes were 2.1 and 2.7, respectively. Theseexperimental results go in line with the analytical estimates. In order to selectthe shape parameter value for the refined node sets we developed a strategybases on controlling the condition number of the interpolation matrix.

We showed that the anisotropic RBF-PUM is suitable for different scalesof ice sheet modeling. The method is more efficient and accurate than thefinite element method in the investigated cases. It becomes even more im-portant for applications with very high aspect ratios since special treatmentsare required for FEM which may lower the accuracy or increase the com-putational complexity. By using the anisotropic RBF-PUM, all the goodfeatures of the isotropic RBF-PUM are preserved without extra cost, whilenew crucial features are gained.

This paper together with [20] provides a base for an RBF framework forice sheet simulations that comprises frozen basal and discontinuous partialslip basal conditions, and moving surface. The method is suitable for vari-ous geometries, even with extreme length/thickness ratios, and capable forapproximating the ice velocity on both structured and unstructured sets ofcomputational nodes, which is an important property for developing adaptivestrategies and taking it further to, say, a grounding line problem [5].

26

Acknowledgements

The authors would like to thank Josefin Ahlkrona, Elisabeth Larsson, Jeremy Levesley,Igor Tominec, Lina von Sydow, and Grady Wright for fruitful discussions. Moreover,the authors are grateful to Elisabeth Larsson and Lina von Sydow for proofreading themanuscript. Cheng Gong was supported by the Swedish Research Council FORMAS grant2013-1600.

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